Monday, August 04, 2008

Analysis in Complexity

An important development in the study of computational complexity has been increased role of analytic methods. Fourier analysis has become an essential tool of the field, playing a critical role in the study of interactive proofs, the computational hardness of approximation problems, and the learnability of Boolean functions. The notion of Gowers uniformity (which was introduced by Gowers to give an analytic proof of Szemeredi's theorem on arithmetic progressions, and whose use can be viewed as "generalized Fourier analysis") has also been recently employed in the context of Probabilistically Checkable Proofs and hardness of approximation. A new paradigm in computational complexity is beginning to emerge, which involves reducing high dimensional discrete problems that arise in the study of Boolean functions to high dimensional continuous problems and then applying analytic methods to the resulting continuous problems.

I started graduate work in complexity right before the algebraic revolution that drove Razborov-Smolensky's circuit lower bounds, Toda's Theorem on the power of the permanent, the power of interactive and probabilistically checkable proofs and much more. But now, two decades later, algebraic techniques are producing diminishing returns and we have seen a growth in using real analysis in complexity as highlighted by this workshop. Avi Wigderson is giving a two-hour survey on "The Power of Partial Derivatives." Hard to have imagined a connection between partial derivatives and complexity.

What about those of us who went into computational complexity because we enjoyed discrete math? Should an old dog like me try to learn new tricks? Ah, the challenge of keeping up with a field that moves in mysterious new ways.

Well, just to help get this very interesting topic underway, it seems to me that these trends are very good news for complexity theory, for the following (mainly heuristic) reasons.

We note that algebraic geometry provides a natural arena for uniting algebraic methods with analytic methods.

Then a physically reasonable approach to both computation and simulation is to specify a state-space---whether classical or quantum---that does accomplishes computational work implicitly, by virtue of the state-space geometry.

This geometrically expresses the main idea of model order reduction. From a discrete point of view, circuit designers apply this same idea whenever they lay out a circuit that accomplishes as much of a computation as possible implicitly.

One virtue of adopting this "GAGA" point of view regarding computation and simulation is that it specifies a fertile mathematical arena within which both algebraic and geometric techniques can be brought to bear simultaneously.

These ideas are particularly well-suited to the study of quantum systems, because it is easier (and more natural) to associate algebraic structures with Kählerian state-spaces than with Riemannian state-spaces.

The recent work of Dowling and Nielsen is an enjoyable and thought-provoking introduction to these ideas.

I definitely hope this thread (and conference) will draw a lot of comments! :)

I started graduate work in complexity right before the algebraic revolution that drove Razborov-Smolensky's circuit lower bounds, .... But now, two decades later, algebraic techniques are producing diminishing returns and we have seen a growth in using real analysis in complexity .... Avi Wigderson is giving a two-hour survey on "The Power of Partial Derivatives." Hard to have imagined a connection between partial derivatives and complexity.

I think you misunderstood something. Wigderson's talk on partial derivatives is precisely an example of the algebraic techniques used in complexity (i.e., partial-derivatives of multivariate polynomials); this was found highly useful in recent years, in contrast to what you said above.

Anon 4 is right, assuming Avi's talk is about the partial derivatives method for arithmetic circuit lower bounds. That method is really algebraic, rather than analytic. Perhaps the scope of the workshop "Analytic Tools in Computational Complexity" is broader than the title implies?